From a different standpoint, it was soon realized that the persistent homology of a filtered finite simplicial complex is simply a particular graded module over a polynomial ring [1]}. Therefore, forgetting about the data and the measurements that yield them, persistence modules can be studied using tools of commutative algebra. In particular, new invariants for persistence modules can be obtained by considering the Betti tablesIn commutative algebra Betti tables are often called (multi-graded) Betti numbers, but we prefer avoiding calling them so to avoid confusion with the Betti numbers of persistent homology groups. of a minimal
free resolution of theirs [2]}.
